
where we define \(P_a = P(\cdot |A=a)\)  and \(P_a^y = P(\cdot |A=a, Y=y)\) , \(a,y \in \lbrace 0,1\rbrace \) . We denote the joint distribution of \(X\)  and \(Y\)  by \(P\) . Similar metrics have also been used in [1]}, [2]}, and [3]}.
One can formulate a penalized optimization problem to regularize the fairness measurement, for instance,
\(\textnormal {(Gap Regularization):}\quad \min _{f} \; \mathbb {E}_{(x,y)\sim P}[\ell (f(x), y)] + \lambda \Delta \text{DP}(f), \) 
